As a small application of the above two theorems, we explain the numerical observations in Ref. [1]}. Under our bilinear rogue solution (REF ), a \(N\) -th order rogue wave exhibits a ring structure when \(a_{2N-1}\)  is large (see Fig. 2 in the next section). In this case, \(m=N-1\) , and \(N_0=N-2\)  from Eq. (). Then, our theory predicts that the center of this \(N\) -th order rogue wave is a \((N-2)\) -th order rogue wave, surrounded by \(N_p=2N-1\)  Peregrine waves that are evenly spaced on a ring due to the \(2\pi /(2m+1)\) -, i.e., \(2\pi /(2N-1)\) -angle rotational symmetry (see Remark 1 above). This is precisely what was observed in [1]}.
